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Syllabus for the Comprehensive Exam in Analysis

Measure Spaces: sigma-algebras, including the sigma-algebras of Borel and Lebesgue measurable sets; measures, including the counting measure and Lebesgue measure; finite and sigma-finite measures; signed measures, complex measures, and product measures

Integration Theory: The Riemann integral; the Lebesgue integral; integration with respect to a measure or signed measure; modes of convergence: in measure, almost surely, and in L^p; convergence theorems: Fatou's lemma, the monotone convergence theorem, and Lebesgue's dominated convergence theorem; product measures and the theorems of Fubini and Tonelli; absolutely continuous measures and the Radon-Niko-dym theorem; singular measures and the Lebesgue decomposition

Topological Spaces: The real number system; metric spaces, including completeness, abstract topological spaces, and the Baire category theorem and its consequences; compact spaces and the Tychonoff theorem